PRICING FORMULAS FOR DOUBLE KNOCK OUT AND BINARY RANGE OPTIONS By: M. Ioffe , G. Ioffe Payoff of Binary Range Option is 1, conditional on Hlow < S(t) < Hup, for any t T or 0, conditional on S(t) Hlow or S(t) Hup, for any t T

(1) (2)

Payoff of Double Knock Out Call option is max(S(t)-K,0) conditional on Hlow < S(t) < Hup, for any t T or 0, conditional on S(t) Hlow or S(t) Hup, for any t T Payoff of Double Knock Out Put option is max(K - S(t),0) conditional on Hlow < S(t) < Hup, for any t T or 0, conditional on S(t) Hlow or S(t) Hup, for any t T where:

t T S(T) Hlow Hup

- any time prior to expiration, - time to expiration, - price of underlying at expiration, - lower boundary, - upper boundary.

In the Black Sholes environment, the risk-neutral asset price process is described by the following stochastic differential equation : dln((S(t)/S) = μ dt + σdW(t) (3) where:

S(t) - asset price at time t, S - current asset price, , - domestic interest rate,

- foreign interest rate for a foreign exchange rate, carry or lease rate dent yield for an index, σ - volatility of asset, W(t) - Wiener process. Let

for commodities or divi-

β = ln(Hlow /S), =ln(Hup / Hlow)/σ, x(t) denotes the diffusion stochastic process which ceases to exist in the points (0, ) and described

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by the following differential equation: dx(t)= μ/σ dt + dW(t). Equation ln(S(t)/S) = σ x(t) + β

(4)

describes all the S(t) that satisfy condition (1). Hence S(0) = S, x(0) = ln( S/ Hlow)/ σ

(5)

The value of a Binary Range or Double Knock Out option with time to expiration T can be written as: V= Where :

(6)

,

= Χ{0, y = x(T), Χ{0, } =

} for Binary Range option 1 if 0 if

or , for Double Knock Out Call option for Double Knock Out Put option

denotes expectation of

.

To solve equation (6) we utilize the fact that the function (7)

where denotes the diffusion stochastic process which ceases to exist at the points 0 and 1 and is given by the following differential equation: , with the initial condition

(8) is described by the following partial differential equation:

(9) with the initial condition

and two constraints:

The solution for equation (9) is

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Given the above equations and initial condition (5) we can rewrite (6) in a closed form. Before we can do so we need to apply rescaling transformations and some additional notation in a form of:

τ = α= Using the above notations the value of a Binary Range Option with time to expiration T can be written as: (10)

B=

The value of a Double Knock Out Call Option with time to expiration T can be written as: C= The value of a Double Knock Out Put Option with time to expiration T can be written as: P= Let Hence

, (11)

where For computation purposes we need to estimate the convergence of series (10). Let Then Let Hence D < 1, (12)

and

Formulas (10), (11) and (12) comprise the close form solution for the value of a Binary Range, Double Knock Out Call and Double Knock Out Put options.

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